Appendix - Three Essays in Applied Microeconomic Theory.

Appendix A: Proof of Existence of the Downstream Equilibrium Under Full Informa-tion

I show that a downstream equilibrium that is characterized in Proposition I.1 must exist by arguing that equations (1.8) and (1.3) are continuous and that they must intersect.

Recall equation (1.3) which describes the high valuation consumer’s threshold:

1 − ¯p = max{0, F (¯c)(1 − v) + (1 − F (¯c))(1 − ¯p) − s}

and let ¯p⁽1.3)(¯c) be the implicit function implied by this equation. By inspection this function is continuous. First note that ¯p⁽1.3)(¯c) > v for all ¯c. This can be seen by contradiction. Rewrite equation (1.3) as

1 − ¯p = max{0, 1 − ¯p + F (¯c)(¯p − v) − s}

If ¯p < v then the left hand side must be larger than the right hand side, hence a contradiction.

Next note that because of the max operator, ¯p⁽1.3)(¯c) ≤ 1 for all ¯c.

Now recall equation (1.8)

c = v − w − ϕ 1 − ϕ

¯ p − v 1 − α(1 − F (¯p − w))

Recall also that ¯p < 1 implies α = 1, else if ¯p = 1 any α ∈ [0, 1] can be used to support an equilibrium. In this sense, α helps the existence argument in that for ¯p = 1, there are many values of ¯c that can satisfy equation (1.8) given a choice of α.

Let equation (1.8) implicitly define the function ¯p⁽1.8⁾(¯c). Continuity once again is obvious here.

First note that ¯p⁽1.8⁾(v −w) = v. Also, note that there exists a ¯c low enough (and possibly negative) so that ¯p⁽1.8⁾(¯c) = 1.

Hence, ¯p⁽1.8⁾(v − w) < ¯p⁽1.3⁾(v − w) and ¯p⁽1.8⁾(¯c) ≥ ¯p⁽1.3⁾(¯c) for some small enough ¯c. Given that ¯p⁽1.8⁾(¯c) and ¯p⁽1.3⁾(¯c) are continuous functions, they must then intersect.

Appendix B: Proof of the Characterization of the Full Information Equilibrium This section provides a proof to Proposition I.2, restated here.

Proposition I.2 The full information equilibrium is characterized by Proposition I.1 and wholesale price w = arg max w · Q(w), as given by equation (1.9). Furthermore, when aggregate demand ϕ and search cost s are sufficiently small, low types are served in equilibrium and high types search until they purchase. When ϕ is sufficiently large no search is induced and low types are excluded.

Proof. The proof proceeds through a series of claims.

Lemma I.5. When ϕ is small enough, the optimal wholesale price chosen by the manufacturer induces sales to low types.

Proof of Lemma I first show that when the proportion of high valuation consumers ϕ is small enough it is feasible for the manufacturer to induce sales to low types. I then show that it is optimal for him to do so.

By equation (1.8)

c = v − w − ϕ 1 − ϕ

¯ p − v 1 − α(1 − F (¯p − w))

≥ v − w − ϕ 1 − ϕ

1 − v F (v − w)

By inspection, the above expression shows that when ϕ is sufficiently small ¯c(w = 0) > 0. In fact, for sufficiently small ϕ there will be a range of wholesale prices that will induce search to low types. Equation (1.9) then shows that that as ϕ is decreased, the quantity sold by setting a wholesale price that serves only high types goes to zero while the quantity that can be sold with a wholesale price that includes low types is bounded strictly above zero. Hence, it is also optimal for the manufacturer to choose a wholesale price that serves low types.

In fact, I prove a stronger statement about the low state. Specifically, if both ϕ and search cost s are sufficiently small, the manufacturer will optimally induce a downstream equilibrium in which α = 1.

Lemma I.6. When ϕ and search cost s are small enough, high type consumers search in equilibrium with probability α = 1.

Proof of Lemma Fix a high type consumer’s strategy ¯p < 1 and α = 1. By Lemma I.5 there exists a ϕ low enough so that the manufacturer chooses a w to induce sales to low types, and let ¯c be the ensuing retailer threshold induced by w. In order for this to be an equilibrium, it must be that ¯p < 1 is a best response to the induced price distribution. By inspection of equation (1.3), for any ¯c and ¯p > v, there exists an s small enough to make the equation hold. Hence ¯p < 1 can be supported in equilibrium and as a result α = 1 is supported.

I have shown that when ϕ is small, the manufacturer sets an equilibrium wholesale price that induces sales to low types. Furthermore, when the search cost s is small, consumers will search in equilibrium. Next I argue that when ϕ is high enough the manufacturer will set a wholesale price that excludes low types.

Lemma I.7. When ϕ is large enough, for any search threshold ¯p > v the manufacturer charges a wholesale price w that excludes low types.

Proof of Lemma By equation (1.1)

c = v − w − ϕ

1 − ϕ(¯p − v)(1 + κ) ≤ v − ϕ 1 − ϕ· s

The second inequality follows from the fact that ¯p − v ≥ s and 1 + κ ≥ 1. The above expression is negative for ϕ large enough, hence the manufacturer will not have the option of inducing sales to low types.

Lastly, if there is an equilibrium in which the manufacturer only serves high types, it must be that there is no search and α = 0. The reason is that if only high types are included, and high types follow a threshold strategy, then only a price of 1 can be supported downstream. If the only price charged is 1 then search is never worthwhile. Along with Lemmas I.5, I.6, and I.7 the proves the proposition.

Appendix C: Proof of Arbitrarily Low Threshold ¯p in the Full Information Equilibrium This section proves that for any small ¯p > 0, there exist small enough ϕ, s, and v so that ¯p is supported in the full information equilibrium. I will show this by first noting that a manufacturer’s profits in any equilibrium are bounded away from zero. Then, I will show that as s and ϕ are reduced toward zero, either ¯p approaches v or the manufacturer’s profit approaches zero, which would contradict the first statement. Lastly, since ¯p can be made arbitrarily close to v, when v is chosen to be small, ¯p will be small as well.

First, recall that by Lemma I.6, the manufacturer chooses to set a wholesale price w to induce sales to low types, i.e. he induces ¯c > 0. It will be useful for this argument to show that as ϕ → 0, the manufacturer’s profit is uniformly bounded strictly above 0.

Claim I.1. For any δ > 0, there exists a low enough ˆϕ so that for any ϕ < ˆϕ, the manufacturer’s equilibrium profit exceeds v/2 · F (v/2) − δ.

Proof of Claim: The manufacturer’s equilibrium profit, given w is the optimally charged wholesale

price, is given by

Consider a manufacturer that charges w = v/2. At this wholesale price, ¯p−w ≥ v+s−w ≥ v/2+s >

0, hence both terms A and B go to zero as ϕ goes to zero. For ϕ low enough, v/2 · (A + B) < δ.

Hence, Π(w, ϕ) ≥ Π(v/2, ϕ) ≥ v/2 · F (v/2) − δ.

Next, I prove by contradiction that as s and ϕ decrease, ¯p − v approaches zero. Suppose to-ward a contradiction that there exists an ε > 0 such that there is some ˆs with the property that for any s < ˆs, in the full information equilibrium ¯p − v > ε. By equation (1.12), this implies that

For small enough ϕ in any equilibrium w < v. By the hypothesis above, this implies that

p − w ≥ ¯p − v ≥ ε for all s < ˆs. Then as s and ϕ both shrink toward zero, the right hand side of the equation above must approach v − w. At the same time, since by hypothesis _F(¯^s_c) > ε for all small s, it must be that ¯c is converging toward zero. Hence, it must be that as s and ϕ shrink toward zero, the equilibrium w approaches v. However, this implies that the manufacturer’s equilibrium profit approaches zero which contradicts the claim above since the manufacturer’s profit has an absolute lower bound strictly above zero.

Hence I have shown that as search cost s and low state demand ϕL diminish toward zero, the equilibrium search threshold ¯p approaches the low types’ valuation v. Thus, there always exists a full information equilibrium where the search threshold is arbitrarily close to zero given parameters ϕ, s, and v are all chosen to be sufficiently small.

Appendix D: Proof of Existence of a No Search Equilibrium When Recommendations are Banned

When recommendations are banned, I show that an equilibrium with no search can be supported when initial belief λ0is sufficiently high.

Proof. In the proposed equilibrium, retailers face the step demand function

q(p) =











1 if p ∈ [0, v]

1 − ϕ if p ∈ (v, 1]

0 if p ∈ (1, ∞)

Retailers use threshold strategy ¯c given by equation (1.1)

c(w, ϕ) = v − w − ϕ

1 − ϕ(1 − v)

Given an equilibrium where consumers do not search, the manufacturer solves

maxw w · Q(w, ϕ) = w ·

(1 − ϕ)F (¯c(w, ϕ)) + ϕF (1 − w)

in each state ϕ, with ¯c(w, ϕ) given above and decreasing in w. Let w(ϕ) be the solution to the above optimization and note that 0 < w(ϕ) < 1.

Claim I.2. When ϕH is large enough, ¯c(ϕH) ≤ 0 and when ϕL is small enough ¯c(ϕL) > 0 in equilibrium.

Proof of Claim This does not follow immediately because in equation (1.1) wholesale prices are endogenous. That ¯c(ϕH) ≤ 0 for a large enough ϕH does follow directly. From the manufac-turer’s optimization it is clear that when ϕL is small enough, setting a w that induces ¯c > 0 is optimal and by (1.1) also feasible for the manufacturer. Hence, when ϕLis small enough ¯cL > 0.

On the supply side I have shown that when consumers follow the strategy of no search then only a price of 1 is charged in the high state and prices v and 1 are charged in the low state. Next, I must

show that no searching is a best response for consumers. For notational clarity, define

wL≡ w(ϕL), wH≡ w(ϕH)

cL≡ ¯c(wL, ϕL), c¯H≡ ¯c(wH, ϕH)

Low valuation consumers expect no prices strictly below v in either state and will either purchase on their first price draw or exit. High valuation consumers will assign a likelihood to the high state conditional on the price they see according to

λ(p) =

Likelihoods at equilibrium prices are computed as the product of the prior likelihood λ0 and the ratio of the probabilities of seeing the price in either state. Prices off the equilibrium path are by assumption ignored by consumers when forming beliefs. Note that likelihood λ translates into belief µ = _λ+1^λ .

A high type consumer whose lowest observed price is p and who holds belief µ has a value function recursively defined by

V (p, µ) = maxn

0, 1 − p, E[V (p^′, µ^′)|p, µ] − so

(1.15)

Claim I.3. There exists a high enough belief ¯µ so that whenever µ > ¯µ, the continuation value to searching E[V (p^′, µ^′)|p, µ] − s < max{0, 1 − p} ∀ p.

Proof of Claim I provide an upper bound for the continuation value to searching:

E[V (p^′, µ^′)|p, µ] − s ≤ µ(max{1 − p, 0}) + (1 − µ)(1 − v) − s

If the state is high, the consumer will not see a price below 1 and the highest payoff she can obtain is to accept p if it is less than 1 else exit. If the state is low, the highest payoff the con-sumer can get is if she observes and accepts price v. For large enough µ, it must then be that

E[V (p^′, µ^′)|p, µ] − s < max{0, 1 − p}.

To restate the claim, for any price p once consumers are convinced enough the state is high they will not search and either purchase or exit.

Claim I.4. There exists a ¯λ so that for any λ0> ¯λ, µ(p) > ¯µ ∀ p.

Proof of Claim The proof follows from equation I have thus shown that for large enough λ0 no price will induce search and this concludes the proof of the lemma.

Optimal Timing of Selection Contests

2.1 Introduction

Going into the 2008 Beijing Olympics hopes were particularly high for American swimmers Michael Phelps and Katie Hoff, both world record holders and slated to compete in numerous events. Yet at the Olympics while Phelps met and exceeded expectations Hoff under-performed. Why the dis-parity in the two athletes’ performances? Putting aside explanations relying on swimmer-specific idiosyncracies, their performances can be understood by considering the athletes’ strategic environ-ment. Swimmers can peak only for a short period and they time their peak for a particular date.

Each could peak closer to the trials to improve their chances of making the team but, by doing so hurt their Olympic performance. Katie Hoff faced stiff domestic competition and was forced to peak closer to the trials; Michael Phelps maintained a comfortable lead over his American rivals and was able to peak closer to the Olympics.¹ For USA Swimming, Katie Hoff’s mediocre performance was avoidable: had they chosen a significantly earlier date for the trials, she would have had the opportunity to recover and peak again at the Olympics.² But holding the trials early would have also come at a cost, since the best swimmer at the trials might no longer still be the best swimmer at the Olympics. In choosing the optimal time to hold the trials, USA Swimming had to balance the accuracy of their selection with the cost of not allowing swimmers sufficient time to recover.

This problem facing USA Swimming is one faced by many organizations: how to time the se-lection of an agent when waiting longer makes the sese-lection more accurate but also more costly.

Take as another example a political party choosing a candidate for a general election. Candidates

1In each event USA Swimming selects the top two performers. Going into the Olypmic trials, Michael Phelps’

average lead over the eventual third place finishers was roughly twice that of Katie Hoff, in percentile terms.

2In fact Australia, another traditional swimming power, held their trials a full year in advance.

will attempt to position themselves on the political spectrum to match the views of their party’s median voter, but in order to win the general election they would be best off at the median of the entire population. Given that shifting one’s political platform takes time, holding a later primary hurts the chances of the eventual primary winner. At the same time, a lengthy primary election can reveal characteristics about a candidate which are independent of his or her platform but are important in determining electability in the general election.³ In this sense a later primary ensures that the winner is the most likely of the group to win the general election. Party organizers must weigh the accuracy benefits of a later primary with the costs of a running a more extreme candidate in the general election.⁴

A similar tradeoff is also found in the workplace. A manager must select which of several ana-lysts to promote to associate. Delaying the promotion decision allows the manager to ascertain which of his analysts is most highly skilled. However, the longer the promotion decision is delayed, the more time is spent by the eventual winner performing the tasks of analyst instead of the more productive tasks of an associate.

In all of these examples, a principal faces a group of agents whose types he does not know and must choose one to perform a task. To make this decision, the principal chooses a time at which all agents compete and uses the results of this competition to make the selection. Agents divert resources from their final task to improve their performance in the competition and the principal must keep this in mind when choosing the timing optimally. The aim of this paper is to explore this timing decision and to ascertain which factors induce the principal to choose a later more accurate contest and which favor an earlier and noisier selection.

We take two approaches to address this question. First we note that from the point of view of the principal, the choice of timing is indirectly a choice of an allocation rule and a corresponding incentive compatible transfer function. That is, for any given selection time, agents choose how many resources to divert to the contest from their final performance, i.e. their transfer, and in an equilibrium conditional on these actions agents will have some probability of winning, i.e. the allocation rule. The choice of timing can be thought of as a blunt tool in the more general

prob-3For example, if candidate is a particularly inspiring public speaker or is has damaging secrets from his or her past.

4The timing of the primary can be broadly understood as the time at which candidates begin to compete, for instance the first televised debate.

lem of choosing an optimal incentive-compatible screening mechanism, and our first approach is to solve that more general problem. Specifically, we consider a setting in which agents are privately informed about their types and simultaneously send costly signals to the principal, who in turn commits to an allocation rule as a function of the signals. The principal’s payoff is the performance of his chosen agent, hence he wants to choose the best agent but suffers that agent’s signaling cost.

We show that ex-ante the principal prefers to allocate not necessarily to agents with the highest type but to those with the lowest hazard ratio, roughly speaking those agents facing the least com-petition. The optimal mechanism is stochastic, that is for some collections of signals the principal will allocate using a lottery between some or all of the agents. While this may seem suboptimal ex-post, a stochastic mechanism lessens the returns to costly signaling and reduces competition ex-ante. Thus, it is best to include some noise in the selection process.

The stochastic optimal allocation rule relies on commitment by the principal and may be difficult to implement in practice. Taken at face value, the mechanism requires the principal to sometimes not select the best agent despite having perfectly inferred all participants’ types through their signals.

The principal cannot always commit to this behavior, just as, for instance, a party chairman may not find it feasible to flip a coin between two candidates when one has received significantly more votes. On the other hand, the party chairman is able to commit to an early primary and this can indirectly infuse a stochastic component into the decision process. Given that a candidate’s types evolves over time, the person chosen early is not necessarily the best one at the time of the general election.⁵

Restricting the principal to making only a timing decision then reduces the set of allocation rules that he can implement, and in our second approach we look for the the optimum in this reduced set. We consider a continuous time environment in which agents privately observe the evolution of their types over time and choose an effort level for the selection contest that comes at the expense of their final performance. We assume that the cost of effort decreases as the time between the selection and the final task grows so that agents are better able to recover given more time. In choosing a later selection time, the principal benefits from the option value of picking the agent with the best shocks but potentially pays the price in higher effort costs. We show that if agents’

ability to divert resources from the final task is unrestricted, agents divert the same amount of

5Or more precisely, if an agent’s type evolves so that it is best at the time of the final task, she may not have necessarily been chosen during the selection contest.

resources from the final task regardless of the timing of the contest. That is, as there is less time to recover between the selection and the final task and effort becomes costlier, all agents adjust their effort to exactly offset this. Hence, it is always optimal to select agents as late as possible.

This result sheds some light on the underlying mechanism and we treat it as a benchmark case.

However, the assumption that agents are unrestricted in how many resources they can divert is strong and violated in many applications. For instance in swimming, even if athletes peak for the trials, they can once again peak for the Olympics given enough time to recover. By choosing a trials date sufficiently early the principal can thus restrict the amount of resources participants can divert. To capture this, we consider a simplified timing model in which the principal chooses between an early and a late contest. The early contest is sufficiently early so that agents will fully recover; hence, they are unable to divert resources from their final performance. The late contest is one in which an agent’s ability to divert resources is unconstrained. We assume that all agents start with the same type at the early contest and since types evolve, each agent’s type is expected to be distributed according to some function F at the time of the late contest. We show that the principal’s payoff to holding a late contest is independent of the mean of F but grows in the

In document Three Essays in Applied Microeconomic Theory. (Page 33-79)